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__FORCETOC__
{{Technical data page}}
=Sensamagic=
The '''sensamagic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the sensamagic comma, [[245/243]].
[[Comma|Comma]]: 245/243


[[7-limit|7-limit]] [[minimax|minimax]]
For a list of rank-2 temperaments, see [[Sensamagic clan]].


[|1 0 0 0>, |0 0 1/5 2/5>, |0 0 1 0>, |0 0 0 1>]
== Sensamagic ==
{{Main| Sensamagic }}


[[Eigenmonzo|Eigenmonzo]]s: 2, 8/7, 5/4
Sensamagic is generated by a perfect fifth and a wide supermajor third of ~[[9/7]], two of which make ~[[5/3]]. Among the good edo tunings are [[87edo]] and [[128edo]], as well as the [[optimal patent val]] [[283edo]].


[[9-limit|9-limit]] minimax
Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 703.7424{{c}}, 440.9020{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.


[|1 0 0 0>, |0 1 0 0>, |0 5/3 2/3 -2/3>, |0 5/3 -1/3 1/3>]
[[Subgroup]]: 2.3.5.7


Eigenmonzos: 2, 4/3, 7/5
[[Comma list]]: 245/243


Lattice basis: 3/2 0.9644 9/7 1.0807
{{Mapping|legend=1| 1 0 0 0 | 0 1 1 2 | 0 0 2 -1 }}
: mapping generators: ~2, ~3, ~9/7


Angle(3/2, 9/7) = 86.5288
[[Mapping to lattice]]: [{{val| 0 1 1 2 }}, {{val| 0 0 2 -1 }}]


Map to lattice: [<0 1 1 2|, <0 0 2 -1|]
Lattice basis:  
: 3/2 length = 0.9644, 9/7 length = 1.0807
: Angle (3/2, 9/7) = 86.5288°


Map: [<1 0 0 0|, <0 1 1 2|, <0 0 2 -1|]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9983{{c}}, ~3/2 = 703.7414{{c}}, ~9/7 = 440.9014{{c}}
: [[error map]]: {{val| -0.002 +1.785 -0.771 -2.248 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7411{{c}}, ~9/7 = 440.9017{{c}}
: error map: {{val| 0.000 +1.786 -0.769 -2.245 }}


Generators: 2, 3, 9/7
[[Minimax tuning]]:  
* [[7-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 0 1/5 2/5 | 0 0 1 0 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.7
* [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 0 5/3 2/3 -2/3 | 0 5/3 -1/3 1/3 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7/5


Edos: [[17edo|,17]] [[19edo|19]], [[22edo|22]], [[24edo|24]], [[27edo|27]], [[41edo|41]], [[46edo|46]], [[60edo|60]], [[68edo|68]], [[87edo|87]], [[109edo|109]], [[128edo|128]], [[169edo|169]], [[177edo|177]], [[237edo|237]]
{{Optimal ET sequence|legend=1| 5, 8d, 14c, 17, 19, 27, 41, 68, 87, 128, 196, 283 }}


[[Badness|Badness]]: 0.000129
[[Badness]] (Sintel): 0.570


[[Projection_pair|Projection pair]]: 5 243/49 to 2.3.7
[[Projection pair]]: 5 243/49 to 2.3.7


===[[Minkowski_blocks|Minkowski blocks]]===
{{Databox|[[Minkowski block]]s|
{2, 3, 7} subgroup
2.3.7 subgroup
* 12: 729/686, 64/63
* 17: 64/63, 19683/19208
* 19: 49/48, 177147/175616
* 22: 64/63, 537824/531441
* 24: 64/63, 15059072/14348907
}}


12: 729/686, 64/63
=== Overview to extensions ===
The second comma in the comma list defines which [[11-limit]] family member we are looking at. Undecimal sensamagic adds [[385/384]], sensawer adds [[441/440]], octarod adds [[100/99]], shrusus adds [[176/175]]. These temperaments use the same generators as sensamagic. Bisector adds [[121/120]] with a half-octave period.


17: 64/63, 19683/19208
Temperaments discussed elsewhere include [[supernatural]] (→ [[Keemic family #Supernatural|Keemic family]]) and [[sensigh]] (→ [[Sengic family #Sensigh|Sengic family]]). The rest are considered below.


19: 49/48, 177147/175616
== Undecimal sensamagic ==
{{Main| Sensamagic }}


22: 64/63, 537824/531441
Undecimal sensamagic tempers out not only [[385/384]], but [[896/891]], making itself a [[strong extension]] of [[parapyth]].


24: 64/63, 15059072/14348907
[[Subgroup]]: 2.3.5.7.11


==Undecimal sensamagic==
[[Comma list]]: 245/243, 385/384
Commas: 245/243, 385/384


[[11-limit|11-limit]] minimax
{{Mapping|legend=1| 1 0 0 0 7 | 0 1 1 2 -2 | 0 0 2 -1 -1 }}


[|1 0 0 0 0>, |21/13 6/13 -1/13 1/13 -3/13>,  
[[Optimal tuning]]s:
|35/13 10/13 7/13 -7/13 -5/13>,
* [[WE]]: ~2 = 1199.9667{{c}}, ~3/2 = 703.7809{{c}}, ~9/7 = 440.9056{{c}}
|35/13 10/13 -6/13 6/13 -5/13>,  
: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
|42/13 -14/13 -2/13 2/13 7/13>]
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.7948{{c}}, ~9/7 = 440.9180{{c}}
: error map: {{val| 0.000 +1.840 -0.683 -2.154 +0.175 }}


Eigenmonzos: 2, 11/9, 7/5
[[Minimax tuning]]:  
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 21/13 6/13 -1/13 1/13 -3/13 }}, {{monzo| 35/13 10/13 7/13 -7/13 -5/13 }}, {{monzo| 35/13 10/13 -6/13 6/13 -5/13 }}, {{monzo| 42/13 -14/13 -2/13 2/13 7/13 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11/9


Map: [<1 0 0 0 7|, <0 1 1 2 -2|, <0 0 2 -1 -1|]
{{Optimal ET sequence|legend=1| 17, 19, 22, 41, 68, 87, 196, 283 }}


Generators: 2, 3, 9/7
[[Badness]] (Sintel): 0.868


EDOs: [[5edo|5]], 17, 19, 22, 41, 68, 87, [[196edo|196]], [[283edo|283]], 607bd, 694bd
[[Projection pair]]s: 5 243/49 11 896/81 to 2.3.7


Badness: 0.000722
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Projection pairs: 5 243/49 11 896/81 to 2.3.7
Comma list: 245/243, 352/351, 364/363


==13-limit==
Mapping: {{mapping| 1 0 0 0 7 12 | 0 1 1 2 -2 -5 | 0 0 2 -1 -1 -1 }}
Commas: 245/243, 352/351, 385/384


Map: [<1 0 0 0 7 12|, <0 1 1 2 -2 -5|, <0 0 2 -1 -1 -1|]
Optimal tunings:  
* WE: ~2 = 1199.9905{{c}}, ~3/2 = 703.7325{{c}}, ~9/7 = 440.9149{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.7381{{c}}, ~9/7 = 440.9184{{c}}


Generators: 2, 3, 9/7
{{Optimal ET sequence|legend=0| 17, 19f, 22, 41, 46, 63, 87, 237, 283 }}


EDOs: 17, 22, 41, 46, 63, 87, 237, 283, 324d, 370bd, 411bd, 607bd, 694bd
Badness (Sintel): 1.12


Badness: 0.00120
== Sensawer ==
[[Subgroup]]: 2.3.5.7.11


=Sensawer=
[[Comma list]]: 245/243, 441/440
Commas: 245/243, 441/440


Map: [<1 0 0 0 -3|, <0 1 1 2 5|, <0 0 2 -1 -4|]
{{Mapping|legend=1| 1 0 0 0 -3 | 0 1 1 2 5 | 0 0 2 -1 -4 }}


EDOs: 41, 87, 302d, 389d, 476bd
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.1654{{c}}, ~3/2 = 703.2870{{c}}, ~9/7 = 441.1967{{c}}
: [[error map]]: {{val| -0.033 +1.793 -0.755 -2.236 +0.048 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.2917{{c}}, ~9/7 = 441.1849{{c}}
: error map: {{val| 0.000 +1.840 -0.683 -2.1554 +0.175 }}


Badness: 0.000796
{{Optimal ET sequence|legend=1| 14c, 19e, 27e, 41, 60e, 87 }}


==13-limit==
[[Badness]] (Sintel): 0.957
Commas: 196/195, 245/243, 352/351


Map: [<1 0 0 0 -3 2|, <0 1 1 2 5 2|, <0 0 2 -1 -4 -4|]
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


EDOs: 41, 46, 87, 522bd
Comma list: 196/195, 245/243, 352/351


Badness: 0.000928
Mapping: {{mapping| 1 0 0 0 -3 2 | 0 1 1 2 5 2 | 0 0 2 -1 -4 -4 }}


=Octarod=
Optimal tunings:
Commas: 100/99, 245/243
* WE: ~2 = 1199.9800{{c}}, ~3/2 = 703.4468{{c}}, ~9/7 = 441.3705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.4494{{c}}, ~9/7 = 441.3758{{c}}


Map: [<1 0 0 0 2|, <0 1 1 2 0|, <0 0 2 -1 4|]
{{Optimal ET sequence|legend=0| 14c, 19e, 27e, 41, 46, 60e, 68e, 87, 522bd }}


EDOs: 19, 22, 41, 63, 104, 109e, 131e, 172e
Badness (Sintel): 0.868


Badness: 0.000581
== Octarod ==
Octarod tempers out [[100/99]] and the interval class of [[11/1|11]] is found as a stack of four ~9/7's. The name ''octarod'' was the former name of the sensamagic comma before being reused for this 11-limit extension, and comes from [[octacot]] and [[rodan]]; it should be noted however that rodan does not temper out 100/99 and therefore does not support this temperament.  


=Shrusus=
[[Subgroup]]: 2.3.5.7.11
Commas: 176/175, 245/243


Map: [<1 0 0 0 -4|, <0 1 1 2 4|, <0 0 2 -1 3|]
[[Comma list]]: 100/99, 245/243


EDOs: 22, 46, 68, 95, 141bc, 163bc, 209bc, 350bc
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 2 0 | 0 0 2 -1 4 }}


Badness: 0.000877
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.2854{{c}}, ~3/2 = 704.6266{{c}}, ~9/7 = 439.2433{{c}}
: [[error map]]: {{val| -0.715 +1.957 -3.915 -0.245 +4.226 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5246{{c}}, ~9/7 = 439.2798{{c}}
: error map: {{val| 0.000 +2.570 -3.230 +0.944 +5.801 }}


==Shrusic==
{{Optimal ET sequence|legend=1| 14c, 19, 22, 27e, 41, 90e, 131e}}*
Commas: 91/90, 176/175, 245/243


Map: [<1 0 0 0 -4 1|, <0 1 1 2 4 1|, <0 0 2 -1 3 3|]
<nowiki/>*[[Optimal patent val]]: [[104edo|104]]


EDOs: 22, 46, 211bcf, 233bcf, 257bcf, 279bcf
[[Badness]] (Sintel): 0.698


Badness: 0.001125
Scales: [[octarod1]], [[octarod2]], [[octarod3]], [[octarod4]], [[octarod5]]


=Sensigh=
== Shrusus ==
Commas: 91/90, 126/125, 169/168
[[Subgroup]]: 2.3.5.7.11


Map: [&lt;1 6 8 11 0 10|, &lt;0 -7 -9 -13 0 -10|, &lt;0 0 0 0 1 0|]
[[Comma list]]: 176/175, 245/243


EDOs: 19, 27, 46, 111df, 157df
{{Mapping|legend=1| 1 0 0 0 -4 | 0 1 1 2 4 | 0 0 2 -1 3 }}


Badness: 0.000939
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1198.9114{{c}}, ~3/2 = 705.7294{{c}}, ~9/7 = 441.7137{{c}}
: [[error map]]: {{val| -1.089 +2.686 +1.754 -1.258 -3.259 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.8402{{c}}, ~9/7 = 442.1064{{c}}
: error map: {{val| 0.000 +3.885 +3.739 +0.748 -1.638 }}


==17-limit==
{{Optimal ET sequence|legend=1| 19e, 22, 27e, 46, 68, 95, 141bc, 163bc }}
Commas: 91/90, 126/125, 154/153, 169/168


Map: [&lt;1 6 8 11 0 10 0|, &lt;0 -7 -9 -13 0 -10 1|, &lt;0 0 0 0 1 0 1|]
[[Badness]] (Sintel): 1.05


EDOs: 19, 27, 46
=== Shrusic ===
Subgroup: 2.3.5.7.11.13


=Bisector=
Comma list: 91/90, 176/175, 245/243
Commas: 121/120, 245/243


Map: [&lt;2 0 0 0 3|, &lt;0 1 1 2 1|, &lt;0 0 2 -1 1|]
Mapping: {{mapping| 1 0 0 0 -4 1 | 0 1 1 2 4 1 | 0 0 2 -1 3 3 }}


EDOs: 22, 46, 68, 82e, 106de, 114, 128e
Optimal tunings:  
* WE: ~2 = 1199.7256{{c}}, ~3/2 = 704.9071{{c}}, ~9/7 = 443.1303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9572{{c}}, ~9/7 = 443.2018{{c}}


Badness: 0.001089
{{Optimal ET sequence|legend=0| 19e, 22, 27e, 46 }}
[[Category:family]]
 
[[Category:overview]]
Badness (Sintel): 1.05
[[Category:planar]]
 
[[Category:sensamagic]]
== Bisector ==
[[Category:theory]]
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 121/120, 245/243
 
{{Mapping|legend=1| 2 0 0 0 3 | 0 1 1 2 1 | 0 0 2 -1 1 }}
: mapping generators: ~77/54, ~3, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 600.3096{{c}}, ~3/2 = 703.4512{{c}}, ~9/7 = 441.3336{{c}}
: [[error map]]: {{val| +0.619 +2.115 +0.424 -2.019 -4.985 }}
* [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 703.5671{{c}}, ~9/7 = 441.2436{{c}}
: error map: {{val| 0.000 +1.612 -0.259 -2.935 -6.507 }}
 
{{Optimal ET sequence|legend=1| 8d, 14c, 22, 38d, 46, 60e, 68, 106de, 128e, 174e }}
 
[[Badness]] (Sintel): 1.31
 
[[Category:Temperament families]]
[[Category:Sensamagic family| ]] <!-- main article -->
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Sensamagic_family"